Edge reinforced random walk (ERRW) and vertex reinforced jump
processes (VRJP) are history dependent stochastic processes,
where the particle tends to come back more often on sites it has
already visited in the past.
For a particular scheme of reinforcement these processes are
random walks in random environment (mixing of reversible Markov chains)
whose mixing measure can be related to a non-linear sigma model introduced
in the context of random matrix models for quantum diffusion.
I will give an overview on these models and explain some recent results.